Error bounds on block Gauss-Seidel solutions of coupled multiphysics problems

• Published in: International journal for numerical methods in engineering Vol. 88; no. 12; pp. 1219 - 1237
• Main Authors: Whiteley, J. P., Gillow, K., Tavener, S. J., Walter, A. C.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 23.12.2011; Wiley
• Subjects: Algebra; block Gauss-Seidel; Blocking; Differential equations; error bound; Errors; Exact sciences and technology; Functionals; Iterative methods; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); multiphysics model; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Solvers; Temperature; Error estimation; block Gauss-Seidel; multiphysics model; Differential equation; Error bound; Non linear effect; Linear functional; Modeling; Gauss Seidel method; Linear model; Multidisciplinary
• ISSN: 0029-5981; 1097-0207; 1097-0207
• DOI: 10.1002/nme.3217

Mathematical models in many fields often consist of coupled sub‐models, each of which describes a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub‐models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non‐linear coupled fluid‐temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration. Copyright © 2011 John Wiley & Sons, Ltd.